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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsymrelsrel | Structured version Visualization version GIF version |
Description: For sets, being an element of the class of symmetric relations (df-symrels 35939) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
Ref | Expression |
---|---|
elsymrelsrel | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrelsrel 35887 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | |
2 | 1 | anbi2d 631 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))) |
3 | elsymrels2 35949 | . 2 ⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | |
4 | dfsymrel2 35945 | . 2 ⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | |
5 | 2, 3, 4 | 3bitr4g 317 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 ◡ccnv 5518 Rel wrel 5524 Rels crels 35615 SymRels csymrels 35624 SymRel wsymrel 35625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-rels 35885 df-ssr 35898 df-syms 35938 df-symrels 35939 df-symrel 35940 |
This theorem is referenced by: elrefsymrelsrel 35967 |
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